`int(x^(c)+c^(x))dx`

If int f(x)dx=psi(x), then int x^(5)f(x^(3))dxint x^(5)f(x^(3))dx(1)/(3)x^(3)psi(x^(3))-3int x^(3)psi(x^(3))dx+C(2)(1)/(3)x^(3)psi(x^(3))-int x^(3)psi(x^(3))dx+C(3)(1)/(3)x^(3)psi(x^(3))-int x^(3)psi(x^(3))dx+C(4)(1)/(3)[x^(3)psi(x^(3))-int x^(2)psi(x^(3))dx]+C

" c) "int e^(x)cos(e^(x))dx

"int(3^(x)e^(3x-2))dx=..........+c

If y=f(x) is a monotonic function in (a,b), then the area bounded by the ordinates at x=a, x=b, y=f(x) and y=f(c)("where "c in (a,b))" is minimum when "c=(a+b)/(2) . "Proof : " A=int_(a)^(c)(f(c)-f(x))dx+int_(c)^(b)(f(c))dx =f(c)(c-a)-int_(a)^(c)(f(x))dx+int_(a)^(b)(f(x))dx-f(c)(b-c) rArr" "A=[2c-(a+b)]f(c)+int_(c)^(b)(f(x))dx-int_(a)^(c)(f(x))dx Differentiating w.r.t. c, we get (dA)/(dc)=[2c-(a+b)]f'(c)+2f(c)+0-f(c)-(f(c)-0) For maxima and minima , (dA)/(dc)=0 rArr" "f'(c)[2c-(a+b)]=0(as f'(c)ne 0) Hence, c=(a+b)/(2) "Also for "clt(a+b)/(2),(dA)/(dc)lt0" and for "cgt(a+b)/(2),(dA)/(dc)gt0 Hence, A is minimum when c=(a+b)/(2) . If the area bounded by f(x)=(x^(3))/(3)-x^(2)+a and the straight lines x=0, x=2, and the x-axis is minimum, then the value of a is

int(a^(x)-b^(x))/(c^(x))dx

Evaluate : int (a^(x)+b^(x))/(c^(x))dx

Property 3:int_(a)^(b)f(x)dx=int_(a)^(c)f(x)dx+int_(c)^(b)f(x)dx